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  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], such as the category of modules over a ring.\nLet $0 \\to A_\\bullet \\xrightarrow{f} B_\\bullet \\xrightarrow{g} C_\\bullet \\to 0$ be a sequence of [[chain complexes|WQ82C0M3QB]] and [[chain maps|HTK9WQZJ1G]] in $\\mathcal{A}$ such that $0 \\to A_n \\to B_n \\to C_n \\to 0$ is a [[short exact sequence|KJKNJMGKZZ]] for every $n$.\nThen there are morphisms $\\partial_n \\colon H_n(C) \\to H_{n-1}(A)$, called *connecting morphisms*, such that the sequence of [[homology|2J4JJJYP22]] objects\n$$\\cdots \\to H_n(A) \\xrightarrow{H_n(f)} H_n(B) \\xrightarrow{H_n(g)} H_n(C) \\xrightarrow{\\partial_n} H_{n-1}(A) \\xrightarrow{H_{n-1}(f)} H_{n-1}(B) \\to \\cdots$$\nis [[exact|JB7ZBFBA9C]].\nThe connecting morphisms are natural with respect to morphisms of such short exact sequences of complexes.\n\n## Remarks\n\nFor modules, $\\partial_n$ is computed as follows.\nLift a cycle $z \\in C_n$ to $y \\in B_n$.\nThen $d y$ maps to $d z = 0$ in $C_{n-1}$, so $d y = f(x)$ for a unique $x \\in A_{n-1}$, which is a cycle, and $\\partial_n [z] = [x]$.\nThe proof applies the [[snake lemma|Q8AYGD93CW]] to the map $\\operatorname{coker} d^A_{n+1} \\to \\ker d^A_{n-1}$ induced by $d$, and the same for $B$ and $C$.\nThe kernels and cokernels of these maps are $H_n$ and $H_{n-1}$, which yields the six-term pieces of the sequence.\nExample: let $A_\\bullet = \\Z$ in degree $0$, let $B_\\bullet$ have $B_1 = B_0 = \\Z$ with $d_1 = 1_{\\Z}$, and let $C_\\bullet = \\Z$ in degree $1$, with the evident inclusion and projection.\nThen $H(B) = 0$, so exactness of $0 = H_1(B) \\to H_1(C) \\to H_0(A) \\to H_0(B) = 0$ makes $\\partial_1 \\colon \\Z \\to \\Z$ an isomorphism.\n\n## Explanations\n\n### L1 human\n\nA short exact sequence of complexes builds the middle complex from a subcomplex and a quotient complex.\nHomology does not keep the sequence short exact, but the theorem says that the failure is controlled.\nThe three homology sequences link up into one long exact sequence, with a connecting map that lowers the degree by one.\n\n### L2 human\n\nThis is the most used tool for computing homology.\nIn topology, the short exact sequences of singular chains for a pair $(X, A)$ give the long exact sequence of a pair.\nFor an open cover by two sets they give the Mayer–Vietoris sequence.\nIn algebra, applying it to a short exact sequence of resolutions gives the long exact sequences of derived functors such as $\\mathrm{Ext}$ and $\\mathrm{Tor}$.\nTogether with the [[five lemma|W6AVEAW2WR]], it lets one prove that a map is a quasi-isomorphism by checking pieces.\n",
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